Modelling a Rhombic triacontahedron
This shape can be modeled using Extra Objects addon or in a manual way.

Extra Objects addon has preset accessible for rhombic triacontahedron. You'll need to activate addon in User Preferences, see more on how to activate / install an addon.

With addon enabled, from Shift+A menu find Mesh > Math function > Regular Solid. Then you'll tweak its parameters:



In Last Operator panel load preset:



Or change parameters for Source, Vertex Truncation and Dual.

To model it manually:


Start with icosphere, 1 subdivision.
Poke faces with Alt+P.. This will create geometry necessary for the middle of rombes.
Select one newly created vertex (from poked ones).
Select similar with Shift+G > Amount of connecting edges. Scale them out along normals a bit.
Select all, convert to quads (Alt+J).





As resulting faces are non-planar (i.e. not all vertices are located on the same imaginary plane), make them planar using Mesh > Clean-Up > Make Planar Faces with around 4 iterations (thanks to Leon Cheung).